On some Hermite–Hadamard inequalities for fractional integrals and their applications
UDC 517.5 We establish some new extensions of Hermite\,--\,Hadamard inequality for fractional integrals and present several applications for the Beta function.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512247588061184 |
|---|---|
| author | Hwang, S.-R. Yeh, S.-Y. Tseng, K.-L. Hwang, S.-R. Yeh, S.-Y. Tseng, K.-L. |
| author_facet | Hwang, S.-R. Yeh, S.-Y. Tseng, K.-L. Hwang, S.-R. Yeh, S.-Y. Tseng, K.-L. |
| author_sort | Hwang, S.-R. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-03-26T11:01:30Z |
| description | UDC 517.5
We establish some new extensions of Hermite\,--\,Hadamard inequality for fractional integrals and present several applications for the Beta function. |
| doi_str_mv | 10.37863/umzh.v72i3.6043 |
| first_indexed | 2026-03-24T03:25:45Z |
| format | Article |
| fulltext |
UDC 517.5
S.-R. Hwang (China Univ. Sci. and Technology, Nankang, Taipei, Taiwan),
S.-Y. Yeh, K.-L. Tseng (Aletheia Univ., Tamsui, New Taipei city, Taiwan)
ON SOME HERMITE – HADAMARD INEQUALITIES
FOR FRACTIONAL INTEGRALS AND THEIR APPLICATIONS
ПРО ДЕЯКI НЕРIВНОСТI ЕРМIТА – АДАМАРА
ДЛЯ ДРОБОВИХ IНТЕГРАЛIВ ТА ЇХ ЗАСТОСУВАННЯ
We establish some new extensions of Hermite – Hadamard inequality for fractional integrals and present several applications
for the Beta function.
Встановлено деякi новi розширення нерiвностi Ермiта – Адамара для дробових iнтегралiв та запропоновано кiлька
застосувань для бета-функцiї.
1. Introduction. Throughout in this paper, let a \leq c < d \leq b in \BbbR with a+ b = c+ d.
The inequality
f
\biggl(
a+ b
2
\biggr)
\leq 1
b - a
b\int
a
f(x) dx \leq f(a) + f(b)
2
(1.1)
which holds for all convex functions f : [a, b] \rightarrow \BbbR , is known in the literature as Hermite – Hadamard
inequality [7].
For some results which generalize, improve, and extend inequality (1.1), see [1 – 6, 8 – 18].
In [4], Dragomir and Agarwal established the following results connected with the second ine-
quality in inequality (1.1).
Theorem A. Let f : [a, b] \rightarrow \BbbR be a differentiable function on (a, b) with a < b. If | f \prime | is
convex on [a, b], then we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f(a) + f(b)
2
- 1
b - a
b\int
a
f(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq b - a
8
\bigl(
| f \prime (a)| + | f \prime (b)|
\bigr)
which is the trapezoid inequality provided | f \prime | is convex on [a, b].
In [12], Kirmaci and Özdemir established the following results connected with the first inequality
in (1.1).
Theorem B. Under the assumptions of Theorem A, we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1
b - a
b\int
a
f(x) dx - f
\biggl(
a+ b
2
\biggr) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq b - a
8
\bigl(
| f \prime (a)| +
\bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr)
which is the midpoint inequality provided | f \prime | is convex on [a, b].
In what follows we recall the following definition [14].
c\bigcirc S.-R. HWANG, S.-Y. YEH, K.-L. TSENG, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3 407
408 S.-R. HWANG, S.-Y. YEH, K.-L. TSENG
Definition 1. Let f \in L1[a, b]. The Riemann – Liouville integrals J\alpha
a+f and J\alpha
b - f of order
\alpha > 0 with a \geq 0 are defined by
J\alpha
a+f(x) =
1
\Gamma (\alpha )
x\int
a
(x - t)\alpha - 1f(t) dt, x > a,
and
J\alpha
b - f(x) =
1
\Gamma (\alpha )
b\int
x
(t - x)\alpha - 1 f(t) dt, x < b,
respectively. Here, \Gamma (\alpha ) is the Gamma function and J0
a+f(x) = J0
b - f(x) = f(x).
In [14], Sarikaya et al. established the following Hermite – Hadamard-type inequalities for frac-
tional integrals.
Theorem C. Let f : [a, b] \rightarrow \BbbR be positive with 0 \leq a < b and f \in L1[a, b]. If f is a convex
function on [a, b], then
f
\biggl(
a+ b
2
\biggr)
\leq \Gamma (\alpha + 1)
2(b - a)\alpha
\bigl[
J\alpha
a+f(b) + J\alpha
b - f(a)
\bigr]
\leq f(a) + f(b)
2
(1.2)
for \alpha > 0.
Theorem D. Under the assumptions of Theorem A, we have the following Hermite – Hadamard-
type inequality for fractional integrals:\bigm| \bigm| \bigm| \bigm| f(a) + f(b)
2
- \Gamma (\alpha + 1)
2(b - a)\alpha
\bigl[
J\alpha
a+f(b) + J\alpha
b - f(a)
\bigr] \bigm| \bigm| \bigm| \bigm| \leq 2\alpha - 1
2\alpha +1(\alpha + 1)
(b - a)
\bigl( \bigm| \bigm| f \prime (a)
\bigm| \bigm| + \bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr)
for \alpha > 0.
In [8], Hwang et al. established the following fractional integral inequality.
Theorem E. Under the assumptions of Theorem A, we have the following Hermite – Hadamard-
type inequality for fractional integrals:\bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
\bigl[
J\alpha
a+f(b) + J\alpha
b - f(a)
\bigr]
- f
\biggl(
a+ b
2
\biggr) \bigm| \bigm| \bigm| \bigm| \leq
\leq b - a
4(\alpha + 1)
\biggl(
\alpha - 1 +
1
2\alpha - 1
\biggr) \bigl(
| f \prime (a)| +
\bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr)
for \alpha > 0.
In [11], Hwang et al. established the following Hermite – Hadamard-type inequalities which are
refinements and similar extensions of Theorems C – E.
Theorem F. Let f : [a, b] \rightarrow \BbbR be a convex function with a < b. Then we have the inequality
f
\biggl(
a+ b
2
\biggr)
\leq
\leq 3\alpha - 1
4\alpha
f
\biggl(
a+ b
2
\biggr)
+
4\alpha - 3\alpha + 1
2 \cdot 4\alpha
\biggl[
f
\biggl(
3a+ b
4
\biggr)
+ f
\biggl(
a+ 3b
4
\biggr) \biggr]
\leq
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
ON SOME HERMITE – HADAMARD INEQUALITIES FOR FRACTIONAL INTEGRALS . . . 409
\leq \Gamma (\alpha + 1)
2(b - a)\alpha
\bigl[
J\alpha
a+f(b) + J\alpha
b - f(a)
\bigr]
\leq
\leq 3\alpha - 1
2 \cdot 4\alpha
\biggl[
f
\biggl(
3a+ b
4
\biggr)
+ f
\biggl(
a+ 3b
4
\biggr) \biggr]
+
4\alpha - 3\alpha + 1
2 \cdot 4\alpha
\bigl[
f(a) + f(b)
\bigr]
\leq
\leq f(a) + f(b)
2
(1.3)
for \alpha > 0.
Theorem G. Under the assumptions of Theorem A, we have the following inequality for frac-
tional integrals: \bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
\bigl[
J\alpha
a+f(b) + J\alpha
b - f(a)
\bigr]
-
-
\biggl\{
3\alpha - 1
2 \cdot 4\alpha
\biggl[
f
\biggl(
3a+ b
4
\biggr)
+ f
\biggl(
a+ 3b
4
\biggr) \biggr]
+
4\alpha - 3\alpha + 1
2 \cdot 4\alpha
\bigl[
f(a) + f(b)
\bigr] \biggr\} \bigm| \bigm| \bigm| \bigm| \leq
\leq K(\alpha )(b - a)
\bigl(
| f \prime (a)| + | f \prime (b)|
\bigr)
,
where
K(\alpha ) :=
2\alpha - 1
2\alpha +1(\alpha + 1)
- 3\alpha - 1
2 \cdot 4\alpha +1
with \alpha > 0.
Theorem H. Under the assumptions of Theorem A, we have the following inequality for frac-
tional integrals: \bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
\bigl[
J\alpha
a+f(b) + J\alpha
b - f(a)
\bigr]
-
-
\biggl\{
3\alpha - 1
4\alpha
f
\biggl(
a+ b
2
\biggr)
+
4\alpha - 3\alpha + 1
2 \cdot 4\alpha
\biggl[
f
\biggl(
3a+ b
4
\biggr)
+ f
\biggl(
a+ 3b
4
\biggr) \biggr] \biggr\} \bigm| \bigm| \bigm| \bigm| \leq
\leq
\biggl(
1
8
- K(\alpha )
\biggr)
(b - a)
\bigl( \bigm| \bigm| f \prime (a)
\bigm| \bigm| + | f \prime (b)|
\bigr)
,
where K(\alpha ) is defined as in Theorem F and \alpha > 0.
Remark 1. 1. The assumptions f : [a, b] \rightarrow \BbbR is positive with 0 \leq a < b in Theorem C can be
weakened as f : [a, b] \rightarrow \BbbR with a < b.
2. In Theorem D, let \alpha = 1. Then Theorem D reduces to Theorem A.
3. In Theorem E, let \alpha = 1. Then Theorem refte reduces to Theorem B.
In this paper, we establish some new extensions of Theorems D – H and present several applica-
tions for the Beta function.
2. New refinements of Hermite – Hadamard-type inequality for fractional integrals. In
this section, we establish some inequalities which refine the inequality (1.2) and generalize the
inequality (1.3).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
410 S.-R. HWANG, S.-Y. YEH, K.-L. TSENG
Theorem 1. Let f : [a, b] \rightarrow \BbbR be a convex function and let a \leq c < d \leq b in \BbbR with
a+ b = c+ d. Then we have the inequality
f
\biggl(
a+ b
2
\biggr)
\leq (b - c)\alpha - (c - a)\alpha
(b - a)\alpha
f
\biggl(
a+ b
2
\biggr)
+
+
(b - a)\alpha + (c - a)\alpha - (b - c)\alpha
2(b - a)\alpha
\bigl[
f(c) + f(d)
\bigr]
\leq
\leq \Gamma (\alpha + 1)
2(b - a)\alpha
\bigl[
J\alpha
a+f(b) + J\alpha
b - f(a)
\bigr]
\leq (b - c)\alpha - (c - a)\alpha
2(b - a)\alpha
\bigl[
f(c) + f(d)
\bigr]
+
+
(b - a)\alpha + (c - a)\alpha - (b - c)\alpha
2(b - a)\alpha
\bigl[
f(a) + f(b)
\bigr]
\leq f(a) + f(b)
2
(2.1)
for \alpha > 0.
Proof. It is easily observed from the convexity of f that the first and last inequalities of (2.1)
hold.
By using simple computation, we have the following identities:
\alpha \Gamma (\alpha )
2(b - a)\alpha
\bigl[
J\alpha
a+f(b) + J\alpha
b - f(a)
\bigr]
=
=
\alpha
2(b - a)\alpha
b\int
a
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr]
f(x) dx =
=
\alpha
2(b - a)\alpha
c\int
a
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr]
[f(x) + f(a+ b - x)] dx+
+
\alpha
2(b - a)\alpha
a+b
2\int
c
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr] \bigl[
f(x) + f(a+ b - x)
\bigr]
dx, (2.2)
\alpha
2(b - a)\alpha
c\int
a
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr]
dx =
(b - a)\alpha + (c - a)\alpha - (b - c)\alpha
2(b - a)\alpha
, (2.3)
\alpha
2(b - a)\alpha
a+b
2\int
c
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr]
dx =
(b - c)\alpha - (c - a)\alpha
2(b - a)\alpha
, (2.4)
c =
a+ b - x - c
a+ b - 2x
x+
c - x
a+ b - 2x
(a+ b - x) =
d - x
c+ d - 2x
x+
c - x
c+ d - 2x
(a+ b - x) (2.5)
and
d =
a+ b - x - d
a+ b - 2x
x+
d - x
a+ b - 2x
(a+ b - x) =
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
ON SOME HERMITE – HADAMARD INEQUALITIES FOR FRACTIONAL INTEGRALS . . . 411
=
c - x
c+ d - 2x
x+
d - x
c+ d - 2x
(a+ b - x), (2.6)
where x \in [a, c] with 0 \leq c - x
c+ d - 2x
,
d - x
c+ d - 2x
\leq 1,
a+ b
2
=
1
2
\bigl[
x+ (a+ b - x)
\bigr]
, (2.7)
where x \in
\biggl[
c,
a+ b
2
\biggr]
,
x =
b - x
b - a
a+
x - a
b - a
b (2.8)
and
a+ b - x =
x - a
b - a
a+
b - x
b - a
b, (2.9)
where x \in [a, c],
x =
d - x
d - c
c+
x - c
d - c
d (2.10)
and
a+ b - x =
d - a - b+ x
d - c
c+
a+ b - x - c
d - c
d
x - c
d - c
c+
d - x
d - c
d, (2.11)
where x \in
\biggl[
c,
a+ b
2
\biggr]
with 0 \leq x - c
d - c
,
d - x
d - c
\leq 1.
Now, by using the above identities and the convexity of f, we have the following inequalities:
(b - a)\alpha + (c - a)\alpha - (b - c)\alpha
2(b - a)\alpha
\bigl[
f(c) + f(d)
\bigr]
=
=
\alpha
2(b - a)\alpha
c\int
a
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr] \bigl[
f(c) + f(d)
\bigr]
dx \leq
\leq \alpha
2(b - a)\alpha
c\int
a
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr]
\times
\times
\biggl[
d - x
c+ d - 2x
f(x) +
c - x
c+ d - 2x
f(a+ b - x)+
+
c - x
c+ d - 2x
f(x) +
d - x
c+ d - 2x
f(a+ b - x)
\biggr]
dx =
=
\alpha
2(b - a)\alpha
c\int
a
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr] \bigl[
f(x) + f(a+ b - x)
\bigr]
dx (2.12)
by identities and (2.5), (2.6),
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
412 S.-R. HWANG, S.-Y. YEH, K.-L. TSENG
(b - c)\alpha - (c - a)\alpha
(b - a)\alpha
f
\biggl(
a+ b
2
\biggr)
=
=
\alpha
2(b - a)\alpha
a+b
2\int
c
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr]
2f
\biggl(
a+ b
2
\biggr)
dx \leq
\leq \alpha
2(b - a)\alpha
a+b
2\int
c
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr] \bigl[
f(x) + f(a+ b - x)
\bigr]
dx (2.13)
by identities (2.4) and (2.7),
\alpha
2(b - a)\alpha
c\int
a
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr] \bigl[
f(x) + f(a+ b - x)
\bigr]
dx \leq
\leq \alpha
2(b - a)\alpha
c\int
a
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr]
\times
\times
\biggl[
b - x
b - a
f(a) +
x - a
b - a
f(b) +
x - a
b - a
f(a) +
b - x
b - a
f(b)
\biggr]
dx =
=
\alpha
\bigl[
f(a) + f(b)
\bigr]
2(b - a)\alpha
c\int
a
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr]
dx =
=
(b - c)\alpha - (c - a)\alpha
2(b - a)\alpha
\bigl[
f(a) + f(b)
\bigr]
(2.14)
by identities (2.3) and (2.8), (2.9),
\alpha
2(b - a)\alpha
a+b
2\int
c
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr] \bigl[
f(x) + f(a+ b - x)
\bigr]
dx \leq
\leq \alpha
2(b - a)\alpha
a+b
2\int
c
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr]
\times
\times
\biggl[
d - x
d - c
f(c) +
x - c
d - c
f(d) +
x - c
d - c
f(c) +
d - x
d - c
f(d)
\biggr]
dx =
=
\alpha
2(b - a)\alpha
a+b
2\int
c
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr] \bigl[
f(c) + f(d)
\bigr]
dx =
=
(b - c)\alpha - (c - a)\alpha
2(b - a)\alpha
\bigl[
f(c) + f(d)
\bigr]
(2.15)
by identities (2.4) and (2.10), (2.11).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
ON SOME HERMITE – HADAMARD INEQUALITIES FOR FRACTIONAL INTEGRALS . . . 413
The second and third inequalities of (2.1) follow from identity (2.2) and inequalities (2.12) –
(2.15).
Theorem 1 is proved.
Remark 2. In Theorem 1, inequality (2.1) refines Hermite – Hadamard-type inequality (1.2).
Corollary 1. In Theorem 1, let c = (1 - \beta )a+\beta b and d = \beta a+(1 - \beta )b with 0 \leq \beta <
1
2
. Then
we have the inequality
f
\biggl(
a+ b
2
\biggr)
\leq [(1 - \beta )\alpha - \beta \alpha ] f
\biggl(
a+ b
2
\biggr)
+
+
\bigl[
1 - (1 - \beta )\alpha + \beta \alpha
\bigr] f((1 - \beta )a+ \beta b) + f(\beta a+ (1 - \beta ) b)
2
\leq
\leq \Gamma (\alpha + 1)
2(b - a)\alpha
\bigl[
J\alpha
a+f(b) + J\alpha
b - f(a)
\bigr]
\leq
\leq
\bigl[
(1 - \beta )\alpha - \beta \alpha
\bigr] f((1 - \beta )a+ \beta b) + f(\beta a+ (1 - \beta )b)
2
+
+
\bigl[
1 - (1 - \beta )\alpha + \beta \alpha
\bigr] f(a) + f(b)
2
\leq f(a) + f(b)
2
.
Remark 3. In Corollary 1, let \beta =
1
4
. Then Corollary 1 reduces to Theorem F.
Remark 4. In Theorem 1, let \alpha = 1. Then we have the inequality
f
\biggl(
a+ b
2
\biggr)
\leq
\leq (1 - 2\beta ) f
\biggl(
a+ b
2
\biggr)
+ \beta
\bigl[
f((1 - \beta )a+ \beta b) + f(\beta a+ (1 - \beta ) b)
\bigr]
\leq
\leq 1
b - a
b\int
a
f(x) dx \leq
\leq 1 - 2\beta
2
\bigl[
f((1 - \beta )a+ \beta b) + f
\bigl(
\beta a+ (1 - \beta )b
\bigr) \bigr]
+ \beta
\bigl[
f(a) + f(b)
\bigr]
\leq
\leq f(a) + f(b)
2
which refines Hermite – Hadamard inequality (1.1).
3. Some extended inequalities for fractional integrals. In this section, we establish two
theorems which are similar extensions of Theorems A – B and D – E.
Theorem 2. Let f : [a, b] \rightarrow \BbbR be a convex function and let a \leq c < d \leq b in \BbbR with
a+ b = c+ d. Then we have the following inequality for fractional integrals:\bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
\bigl[
J\alpha
a+f(b) + J\alpha
b - f(a)
\bigr]
-
-
\biggl(
(b - c)\alpha - (c - a)\alpha
2(b - a)\alpha
\bigl[
f(c) + f(d)
\bigr]
+
(b - a)\alpha - (b - c)\alpha + (c - a)\alpha
2(b - a)\alpha
[f(a) + f(b)]
\biggr) \bigm| \bigm| \bigm| \bigm| \leq
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
414 S.-R. HWANG, S.-Y. YEH, K.-L. TSENG
\leq H\alpha (c)(b - a)
\bigl(
| f \prime (a)| + | f \prime (b)|
\bigr)
, (3.1)
where
H\alpha (c) :=
2\alpha - 1
2\alpha +1(\alpha + 1)
- c - a
2(b - a)
\biggl[ \biggl(
b - c
b - a
\biggr) \alpha
-
\biggl(
c - a
b - a
\biggr) \alpha \biggr]
with \alpha > 0.
Proof. Define
h1(x) =
\left\{
(b - x)\alpha - (x - a)\alpha - (b - c)\alpha + (c - a)\alpha , x \in [a, c),
(b - x)\alpha - (x - a)\alpha , x \in [c, d),
(b - x)\alpha - (x - a)\alpha + (b - c)\alpha - (c - a)\alpha , x \in [d, b].
By using the integration by parts, we have the following identities:
1
2(b - a)\alpha
b\int
a
h1(x)f
\prime (x) dx =
=
\alpha
2(b - a)\alpha
b\int
a
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr]
f(x) dx -
-
\biggl\{
(b - c)\alpha - (c - a)\alpha
2(b - a)\alpha
\bigl[
f(c) + f(d)
\bigr]
+
+
(b - a)\alpha - (b - c)\alpha + (c - a)\alpha
2(b - a)\alpha
\bigl[
f(a) + f(b)
\bigr] \biggr\}
=
=
\alpha \Gamma (\alpha )
2(b - a)\alpha
\bigl[
J\alpha
a+f(b) + J\alpha
b - f(a)
\bigr]
-
\biggl\{
(b - c)\alpha - (c - a)\alpha
2(b - a)\alpha
\bigl[
f(c) + f(d)
\bigr]
+
+
(b - a)\alpha - (b - c)\alpha + (c - a)\alpha
2(b - a)\alpha
[f(a) + f(b)]
\biggr\}
=
=
\Gamma (\alpha + 1)
2(b - a)\alpha
\bigl[
J\alpha
a+f(b) + J\alpha
b - f(a)
\bigr]
-
\biggl\{
(b - c)\alpha - (c - a)\alpha
2(b - a)\alpha
\bigl[
f(c) + f(d)
\bigr]
+
+
(b - a)\alpha - (b - c)\alpha + (c - a)\alpha
2(b - a)\alpha
\bigl[
f(a) + f(b)
\bigr] \biggr\}
, (3.2)
c\int
a
\bigl[
(b - x)\alpha - (x - a)\alpha - (b - c)\alpha + (c - a)\alpha
\bigr] b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| dx+
+
b\int
d
\bigl[
(x - a)\alpha - (b - x)\alpha - (b - c)\alpha + (c - a)\alpha
\bigr] b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| dx =
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ON SOME HERMITE – HADAMARD INEQUALITIES FOR FRACTIONAL INTEGRALS . . . 415
=
c\int
a
\bigl[
(b - x)\alpha - (x - a)\alpha - (b - c)\alpha + (c - a)\alpha
\bigr] b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| dx+
+
c\int
a
\bigl[
(b - x)\alpha - (x - a)\alpha - (b - c)\alpha + (c - a)\alpha
\bigr] x - a
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| dx =
=
\bigm| \bigm| f \prime (a)
\bigm| \bigm| c\int
a
\bigl[
(b - x)\alpha - (x - a)\alpha - (b - c)\alpha + (c - a)\alpha
\bigr]
dx = P1, (3.3)
where
P1 :=
\bigm| \bigm| f \prime (a)
\bigm| \bigm| \biggl\{ 1
\alpha + 1
\bigl[
(b - a)\alpha +1 - (b - c)\alpha +1 - (c - a)\alpha +1
\bigr]
-
- (c - a)
\bigl[
(b - c)\alpha - (c - a)\alpha
\bigr] \biggr\}
,
(3.4)
c\int
a
\bigl[
(b - x)\alpha - (x - a)\alpha - (b - c)\alpha + (c - a)\alpha
\bigr] x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| dx+
+
b\int
d
\bigl[
(x - a)\alpha - (b - x)\alpha - (b - c)\alpha + (c - a)\alpha
\bigr] x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| dx =
=
c\int
a
\bigl[
(b - x)\alpha - (x - a)\alpha - (b - c)\alpha + (c - a)\alpha
\bigr] x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| dx+
+
c\int
a
\bigl[
(b - x)\alpha - (x - a)\alpha - (b - c)\alpha + (c - a)\alpha
\bigr] b - x
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| dx =
=
\bigm| \bigm| f \prime (b)
\bigm| \bigm| c\int
a
\bigl[
(b - x)\alpha - (x - a)\alpha - (b - c)\alpha + (c - a)\alpha
\bigr]
dx = P2,
where
P2 :=
\bigm| \bigm| f \prime (b)
\bigm| \bigm| \biggl\{ 1
\alpha + 1
\bigl[
(b - a)\alpha +1 - (b - c)\alpha +1 - (c - a)\alpha +1
\bigr]
-
- (c - a)
\bigl[
(b - c)\alpha - (c - a)\alpha
\bigr] \biggr\}
,
(3.5)
a+b
2\int
c
\bigl[
(b - x)\alpha - (x - a)\alpha
\bigr] b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| dx+
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416 S.-R. HWANG, S.-Y. YEH, K.-L. TSENG
+
d\int
a+b
2
[(x - a)\alpha - (b - x)\alpha ]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| dx =
=
a+b
2\int
c
\bigl[
(b - x)\alpha - (x - a)\alpha
\bigr] b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| dx+
+
a+b
2\int
c
\bigl[
(b - x)\alpha - (x - a)\alpha
\bigr] x - a
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| dx =
=
\bigm| \bigm| f \prime (a)
\bigm| \bigm| a+b
2\int
c
\bigl[
(b - x)\alpha - (x - a)\alpha
\bigr]
dx = P3,
where
P3 :=
\bigm| \bigm| f \prime (a)
\bigm| \bigm|
\alpha + 1
\biggl[
(b - c)\alpha +1 + (c - a)\alpha +1 - (b - a)\alpha +1
2\alpha
\biggr]
,
(3.6)
a+b
2\int
c
\bigl[
(b - x)\alpha - (x - a)\alpha
\bigr] x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| dx+
+
d\int
a+b
2
[(x - a)\alpha - (b - x)\alpha ]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| dx =
=
a+b
2\int
c
\bigl[
(b - x)\alpha - (x - a)\alpha
\bigr] x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| dx+
+
a+b
2\int
c
\bigl[
(b - x)\alpha - (x - a)\alpha
\bigr] b - x
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| dx =
=
\bigm| \bigm| f \prime (b)
\bigm| \bigm| a+b
2\int
3a+b
4
\bigl[
(b - x)\alpha - (x - a)\alpha
\bigr]
dx = P4,
where
P4 :=
\bigm| \bigm| f \prime (b)
\bigm| \bigm|
\alpha + 1
\biggl[
(b - c)\alpha +1 + (c - a)\alpha +1 - (b - a)\alpha +1
2\alpha
\biggr]
.
Now, by using simple computation and identities (2.8) and (3.3) – (3.6), we have the inequality
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
ON SOME HERMITE – HADAMARD INEQUALITIES FOR FRACTIONAL INTEGRALS . . . 417\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1
2(b - a)\alpha
b\int
a
h1(x)f
\prime (x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq 1
2(b - a)\alpha
b\int
a
| h1(x)|
\bigm| \bigm| f \prime (x)
\bigm| \bigm| dx =
=
1
2(b - a)\alpha
c\int
a
[(b - x)\alpha - (x - a)\alpha - (b - c)\alpha + (c - a)\alpha ]
\bigm| \bigm| f \prime (x)
\bigm| \bigm| dx+
+
1
2(b - a)\alpha
a+b
2\int
c
\bigl[
(b - x)\alpha - (x - a)\alpha
\bigr] \bigm| \bigm| f \prime (x)
\bigm| \bigm| dx+
+
1
2(b - a)\alpha
+
d\int
a+b
2
[(x - a)\alpha - (b - x)\alpha ]
\bigm| \bigm| f \prime (x)
\bigm| \bigm| dx+
+
1
2(b - a)\alpha
b\int
d
\bigl[
(x - a)\alpha - (b - x)\alpha - (b - c)\alpha + (c - a)\alpha
\bigr] \bigm| \bigm| f \prime (x)
\bigm| \bigm| dx \leq
\leq P1 + P2 + P3 + P4
2(b - a)\alpha
=
(b - a)
\bigl( \bigm| \bigm| f \prime (a)
\bigm| \bigm| + \bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr)
2
\times
\times
\Biggl\{
1
\alpha + 1
\Biggl[
1 -
\biggl(
b - c
b - a
\biggr) \alpha +1
-
\biggl(
c - a
b - a
\biggr) \alpha +1
\Biggr]
- c - a
b - a
\biggl[ \biggl(
b - c
b - a
\biggr) \alpha
-
\biggl(
c - a
b - a
\biggr) \alpha \biggr] \Biggr\}
+
+
(b - a) (| f \prime (a)| + | f \prime (b)| )
2(\alpha + 1)
\Biggl( \biggl(
b - c
b - a
\biggr) \alpha +1
+
\biggl(
c - a
b - a
\biggr) \alpha +1
- 1
2\alpha
\Biggr)
=
= H\alpha (c)(b - a)
\bigl( \bigm| \bigm| f \prime (a)
\bigm| \bigm| + \bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr) . (3.7)
Inequality (3.1) follows from identity (3.2) and inequality (3.7).
Theorem 2 is proved.
Remark 5. In Theorem 2, let c = a. Then Theorem 2 reduces to Theorem D.
Corollary 2. In Theorem 2, let c = (1 - \beta )a+\beta b and d = \beta a+(1 - \beta )b with 0 \leq \beta <
1
2
. Then
we have the inequality\bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
\bigl[
J\alpha
a+f(b) + J\alpha
b - f(a)
\bigr]
-
\biggl\{
(1 - \beta )\alpha - \beta \alpha
2
\bigl[
f
\bigl(
(1 - \beta )a+ \beta b
\bigr)
+
+f(\beta a+ (1 - \beta )b)] +
1 - (1 - \beta )\alpha + \beta \alpha
2
\bigl[
f(a) + f(b)
\bigr] \biggr\} \bigm| \bigm| \bigm| \bigm| \leq
\leq M\alpha (\beta )(b - a)
\bigl( \bigm| \bigm| f \prime (a)
\bigm| \bigm| + \bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr) ,
where
M\alpha (\beta ) :=
2\alpha - 1
2\alpha +1 (\alpha + 1)
- \beta
2
\bigl[
(1 - \beta )\alpha - \beta \alpha
\bigr]
with \alpha > 0.
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418 S.-R. HWANG, S.-Y. YEH, K.-L. TSENG
Remark 6. In Corollary 2, let \beta =
1
4
. Then Corollary 2 reduces to Theorem G.
Remark 7. In Corollary 2, let \alpha = 1. Then we have Hermite – Hadamard-type inequality\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1
b - a
b\int
a
f(x) dx -
\biggl\{
1 - 2\beta
2
\bigl[
f((1 - \beta )a+ \beta b) + f(\beta a+ (1 - \beta )b)
\bigr]
+ \beta
\bigl[
f(a) + f(b)
\bigr] \biggr\} \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\biggl[
1
8
- \beta
\biggl(
1
2
- \beta
\biggr) \biggr]
(b - a)
\bigl( \bigm| \bigm| f \prime (a)
\bigm| \bigm| + \bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr) .
Remark 8. In Remark 6, let \beta = 0. Then Remark 6 reduces to Theorem A.
Theorem 3. Let f : [a, b] \rightarrow \BbbR be a convex function and a \leq c < d \leq b in \BbbR with a+b = c+d.
Then we have the following inequality for fractional integrals:\bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
\bigl[
J\alpha
a+f(b) + J\alpha
b - f(a)
\bigr]
-
\biggl\{
(b - c)\alpha - (c - a)\alpha
(b - a)\alpha
f
\biggl(
a+ b
2
\biggr)
+
+
(b - a)\alpha + (c - a)\alpha - (b - c)\alpha
2(b - a)\alpha
\bigl[
f(c) + f(d)
\bigr] \biggr\} \bigm| \bigm| \bigm| \bigm| \leq I\alpha (c)(b - a)
\bigl(
| f \prime (a)| +
\bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr) , (3.8)
where
I\alpha (c) :=
c - a
2(b - a)
+
\biggl[
1
4
- c - a
2(b - a)
\biggr] \biggl[ \biggl(
b - c
b - a
\biggr) \alpha
-
\biggl(
c - a
b - a
\biggr) \alpha \biggr]
- 2\alpha - 1
2\alpha +1(\alpha + 1)
with \alpha > 0.
Proof. Define
h2(x) =
\left\{
(b - x)\alpha - (x - a)\alpha - (b - a)\alpha , x \in [a, c),
(b - x)\alpha - (x - a)\alpha - (b - c)\alpha + (c - a)\alpha , x \in
\biggl[
c,
a+ b
2
\biggr)
,
(b - x)\alpha - (x - a)\alpha + (b - c)\alpha - (c - a)\alpha , x \in
\biggl[
a+ b
2
, d
\biggr)
,
(b - x)\alpha - (x - a)\alpha + (b - a)\alpha , x \in [d, b].
By using the integration by parts, we have the following identities:
1
2(b - a)\alpha
b\int
a
h2(x)f
\prime (x) dx =
\alpha
2(b - a)\alpha
b\int
a
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr]
f(x) dx -
-
\biggl\{
(b - c)\alpha - (c - a)\alpha
(b - a)\alpha
f
\biggl(
a+ b
2
\biggr)
+
+
(b - a)\alpha + (c - a)\alpha - (b - c)\alpha
2(b - a)\alpha
\bigl[
f(c) + f(d)
\bigr] \biggr\}
=
=
\alpha \Gamma (\alpha )
2(b - a)\alpha
\bigl[
J\alpha
a+f(b) + J\alpha
b - f(a)
\bigr]
-
\biggl\{
(b - c)\alpha - (c - a)\alpha
(b - a)\alpha
f
\biggl(
a+ b
2
\biggr)
+
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ON SOME HERMITE – HADAMARD INEQUALITIES FOR FRACTIONAL INTEGRALS . . . 419
+
(b - a)\alpha + (c - a)\alpha - (b - c)\alpha
2(b - a)\alpha
\bigl[
f(c) + f(d)
\bigr] \biggr\}
=
=
\Gamma (\alpha + 1)
2(b - a)\alpha
\bigl[
J\alpha
a+f(b) + J\alpha
b - f(a)
\bigr]
-
\biggl\{
(b - c)\alpha - (c - a)\alpha
(b - a)\alpha
f
\biggl(
a+ b
2
\biggr)
+
+
(b - a)\alpha + (c - a)\alpha - (b - c)\alpha
2(b - a)\alpha
\bigl[
f(c) + f(d)
\bigr] \biggr\}
, (3.9)
c\int
a
\bigl[
(x - a)\alpha - (b - x)\alpha + (b - a)\alpha
\bigr] b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| dx+
+
b\int
d
\bigl[
(b - x)\alpha - (x - a)\alpha + (b - a)\alpha
\bigr] b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| dx =
=
c\int
a
\bigl[
(x - a)\alpha - (b - x)\alpha + (b - a)\alpha
\bigr] b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| dx+
+
c\int
a
\bigl[
(x - a)\alpha - (b - x)\alpha + (b - a)\alpha
\bigr] x - a
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| dx =
=
\bigm| \bigm| f \prime (a)
\bigm| \bigm| c\int
a
\bigl[
(x - a)\alpha - (b - x)\alpha + (b - a)\alpha
\bigr]
dx = Q1, (3.10)
where
Q1 :=
\bigm| \bigm| f \prime (a)
\bigm| \bigm| \biggl\{ 1
\alpha + 1
\bigl[
(b - c)\alpha +1 + (c - a)\alpha +1 - (b - a)\alpha +1
\bigr]
+ (c - a)(b - a)\alpha
\biggr\}
,
(3.11)
c\int
a
\bigl[
(x - a)\alpha - (b - x)\alpha + (b - a)\alpha
\bigr] x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| dx+
+
b\int
d
\bigl[
(b - x)\alpha - (x - a)\alpha + (b - a)\alpha
\bigr] x - a
b - a
| f \prime (b)| dx =
=
c\int
a
\bigl[
(x - a)\alpha - (b - x)\alpha + (b - a)\alpha
\bigr] x - a
b - a
| f \prime (b)| dx+
+
c\int
a
\bigl[
(x - a)\alpha - (b - x)\alpha + (b - a)\alpha
\bigr] b - x
b - a
| f \prime (b)| dx =
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420 S.-R. HWANG, S.-Y. YEH, K.-L. TSENG
=
\bigm| \bigm| f \prime (b)
\bigm| \bigm| c\int
a
\bigl[
(x - a)\alpha - (b - x)\alpha + (b - a)\alpha
\bigr]
dx = Q2,
where
Q2 :=
\bigm| \bigm| f \prime (b)
\bigm| \bigm| \biggl\{ 1
\alpha + 1
\bigl[
(b - c)\alpha +1 + (c - a)\alpha +1 - (b - a)\alpha +1
\bigr]
+ (c - a)(b - a)\alpha
\biggr\}
,
(3.12)
a+b
2\int
c
\bigl[
(x - a)\alpha - (b - x)\alpha + (b - c)\alpha - (c - a)\alpha
\bigr] b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| dx+
+
d\int
a+b
2
\bigl[
(b - x)\alpha - (x - a)\alpha + (b - c)\alpha - (c - a)\alpha
\bigr] b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| dx =
=
a+b
2\int
c
\bigl[
(x - a)\alpha - (b - x)\alpha + (b - c)\alpha - (c - a)\alpha
\bigr] b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| dx+
+
a+b
2\int
c
\bigl[
(x - a)\alpha - (b - x)\alpha + (b - c)\alpha - (c - a)\alpha
\bigr] x - a
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| dx =
=
\bigm| \bigm| f \prime (a)
\bigm| \bigm| a+b
2\int
3a+b
4
\bigl[
(x - a)\alpha - (b - x)\alpha + (b - c)\alpha - (c - a)\alpha
\bigr]
dx = Q3,
where
Q3 :=
\bigm| \bigm| f \prime (a)
\bigm| \bigm| \biggl\{ 1
\alpha + 1
\biggl[
(b - a)\alpha +1
2\alpha
- (b - c)\alpha +1 - (c - a)\alpha +1
\biggr]
+
+
a+ b - 2c
2
\bigl[
(b - c)\alpha - (c - a)\alpha
\bigr] \biggr\}
,
(3.13)
a+b
2\int
c
\bigl[
(x - a)\alpha - (b - x)\alpha + (b - c)\alpha - (c - a)\alpha
\bigr] x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| dx+
+
d\int
a+b
2
\bigl[
(b - x)\alpha - (x - a)\alpha + (b - c)\alpha - (c - a)\alpha
\bigr] x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| dx =
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ON SOME HERMITE – HADAMARD INEQUALITIES FOR FRACTIONAL INTEGRALS . . . 421
=
a+b
2\int
c
\bigl[
(x - a)\alpha - (b - x)\alpha + (b - c)\alpha - (c - a)\alpha
\bigr] x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| dx+
+
a+b
2\int
c
\bigl[
(x - a)\alpha - (b - x)\alpha + (b - c)\alpha - (c - a)\alpha
\bigr] b - x
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| dx =
=
\bigm| \bigm| f \prime (b)
\bigm| \bigm| a+b
2\int
c
\bigl[
(x - a)\alpha - (b - x)\alpha + (b - c)\alpha - (c - a)\alpha
\bigr]
dx = Q4,
where
Q4 :=
\bigm| \bigm| f \prime (b)
\bigm| \bigm| \biggl\{ 1
\alpha + 1
\biggl[
(b - a)\alpha +1
2\alpha
- (b - c)\alpha +1 - (c - a)\alpha +1
\biggr]
+
+
a+ b - 2c
2
\bigl[
(b - c)\alpha - (c - a)\alpha
\bigr] \biggr\}
.
Now, by using simple computation and identities (2.8) and (3.10) – (3.13), we have the inequality\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1
2(b - a)\alpha
b\int
a
h2(x)f
\prime (x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq 1
2(b - a)\alpha
b\int
a
| h2(x)|
\bigm| \bigm| f \prime (x)
\bigm| \bigm| dx =
=
1
2(b - a)\alpha
c\int
a
[(x - a)\alpha - (b - x)\alpha + (b - a)\alpha ]
\bigm| \bigm| f \prime (x)
\bigm| \bigm| dx+
+
1
2(b - a)\alpha
a+b
2\int
c
[(x - a)\alpha - (b - x)\alpha + (b - c)\alpha - (c - a)\alpha ]
\bigm| \bigm| f \prime (x)
\bigm| \bigm| dx+
+
1
2(b - a)\alpha
+
d\int
a+b
2
[(b - x)\alpha - (x - a)\alpha + (b - c)\alpha - (c - a)\alpha ]
\bigm| \bigm| f \prime (x)
\bigm| \bigm| dx+
+
1
2(b - a)\alpha
b\int
d
[(b - x)\alpha - (x - a)\alpha + (b - a)\alpha ]
\bigm| \bigm| f \prime (x)
\bigm| \bigm| dx \leq Q1 +Q2 +Q3 +Q4
2(b - a)\alpha
=
=
(b - a)
\bigl( \bigm| \bigm| f \prime (a)
\bigm| \bigm| + \bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr)
2
\Biggl\{
1
\alpha + 1
\Biggl[ \biggl(
b - c
b - a
\biggr) \alpha +1
++
\biggl(
c - a
b - a
\biggr) \alpha +1
- 1
\Biggr]
+
c - a
b - a
\Biggr\}
+
+
(b - a)
\bigl( \bigm| \bigm| f \prime (a)
\bigm| \bigm| + \bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr)
2
\Biggl\{
1
\alpha + 1
\Biggl[
1
2\alpha
-
\biggl(
b - c
b - a
\biggr) \alpha +1
-
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
422 S.-R. HWANG, S.-Y. YEH, K.-L. TSENG
-
\biggl(
c - a
b - a
\biggr) \alpha +1
\Biggr]
+
\biggl(
1
2
- b - c
b - a
\biggr) \biggl[ \biggl(
b - c
b - a
\biggr) \alpha
-
\biggl(
c - a
b - a
\biggr) \alpha \biggr] \Biggr\}
=
= I\alpha (c)(b - a)
\Bigl( \bigm| \bigm| f \prime (a)
\bigm| \bigm| + \bigm| \bigm| f \prime (b)
\bigm| \bigm| \Bigr) . (3.14)
Inequality (3.8) follows from identity (3.9) and inequality (3.14).
Theorem 3 is proved.
Remark 9. In Theorem 3, let c = a. Then Theorem 3 reduces to Theorem E.
Corollary 3. In Theorem 3, let c = (1 - \beta )a+\beta b and d = \beta a+(1 - \beta )b with 0 \leq \beta <
1
2
. Then
we have the inequality\bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
\bigl[
J\alpha
a+f(b) + J\alpha
b - f(a)
\bigr]
-
\biggl\{ \bigl[
(1 - \beta )\alpha - \beta \alpha
\bigr]
f
\biggl(
a+ b
2
\biggr)
+
+
1 - (1 - \beta )\alpha + \beta \alpha
2
\bigl[
f
\bigl(
(1 - \beta )a+ \beta b
\bigr)
+ f(\beta a+ (1 - \beta )b)
\bigr] \biggr\} \bigm| \bigm| \bigm| \bigm| \leq
\leq N\alpha (\beta )(b - a)
\bigl( \bigm| \bigm| f \prime (a)
\bigm| \bigm| + \bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr) ,
where
N\alpha (\beta ) :=
\beta
2
+
\biggl(
1
4
- \beta
2
\biggr) \bigl[
(1 - \beta )\alpha - \beta \alpha
\bigr]
- 2\alpha - 1
2\alpha +1(\alpha + 1)
with \alpha > 0.
Remark 10. In Corollary 2, let \beta =
1
4
. Then Corollary 2 reduces to Theorem H.
Remark 11. In Corollary 3, let \alpha = 1. Then we have the Hermite – Hadamard-type inequality\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1
b - a
b\int
a
f(x) dx -
\biggl\{
(1 - 2\beta )f
\biggl(
a+ b
2
\biggr)
+ \beta
\bigl[
f((1 - \beta )a+ \beta b) + f(\beta a+ (1 - \beta )b)
\bigr] \biggr\} \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\biggl[
1
8
- \beta
\biggl(
1
2
- \beta
\biggr) \biggr]
(b - a)
\bigl( \bigm| \bigm| f \prime (a)
\bigm| \bigm| + \bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr) .
Remark 12. In Remark 6, let \beta = 0. Then Remark 11 reduces to Theorem B.
Remark 13. In Corollaries 2 and 3, let \beta =
1
4
. Then we obtain the following inequalities:
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1
b - a
b\int
a
f(x) dx - 1
4
\biggl[
f(a) + f
\biggl(
3a+ b
4
\biggr)
+ f
\biggl(
a+ 3b
4
\biggr)
+ f(b)
\biggr] \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq b - a
16
\bigl( \bigm| \bigm| f \prime (a)
\bigm| \bigm| + \bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr)
and \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1
b - a
b\int
a
f(x) dx - 1
4
\biggl[
f
\biggl(
3a+ b
4
\biggr)
+ 2f
\biggl(
a+ b
2
\biggr)
+ f
\biggl(
a+ 3b
4
\biggr) \biggr] \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
ON SOME HERMITE – HADAMARD INEQUALITIES FOR FRACTIONAL INTEGRALS . . . 423
\leq b - a
16
\bigl( \bigm| \bigm| f \prime (a)
\bigm| \bigm| + \bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr)
which are similar extensions of Theorems A and B.
4. Applications for the Beta functions. Throughout this section, let \alpha > 0, a = 0, b = 1,
\Gamma (\alpha ) be the Gamma function andf(x) = x\rho - 1 (\rho > 1, x \in [0, 1]).
Let us recall the Beta function
B(p, q) =
1\int
0
xp - 1 (1 - x)q - 1 dx, p, q > 0.
Remark 14. In Sections 2 and 3, we obtain
\Gamma (\alpha + 1)
2(b - a)\alpha
J\alpha
a+f(b) =
\alpha
2
1\int
0
(1 - x)\alpha - 1x\rho - 1dx =
\alpha
2
B(\rho , \alpha )
and
\Gamma (\alpha + 1)
2(b - a)\alpha
J\alpha
b - f(a) =
\alpha
2
1\int
0
x\alpha +\rho - 2dx =
\alpha
2(\alpha + \rho - 1)
.
By using Corollaries 1 – 3 and Remark 14, we have the following propositions.
Proposition 1. Let \rho \geq 2, 0 \leq \beta <
1
2
, c = \beta and d = 1 - \beta in Corollary 1. Then the following
inequality holds:
1
2\rho - 1
(1 - \beta )\alpha - \beta \alpha
2\rho - 1
+
1 - (1 - \beta )\alpha + \beta \alpha
2
\bigl[
(1 - \beta )\rho - 1 + \beta \rho - 1
\bigr]
\leq
\leq \alpha
2
B(\rho , \alpha ) +
\alpha
2(\alpha + \rho - 1)
\leq
\leq (1 - \beta )\alpha - \beta \alpha
2
\bigl[
(1 - \beta )\rho - 1 + \beta \rho - 1
\bigr]
+
1 - (1 - \beta )\alpha + \beta \alpha
2
\leq 1
2
.
Proposition 2. Let \rho \geq 3, 0 \leq \beta <
1
2
, c = \beta and d = 1 - \beta in Corollary 2. Then, on the basis
of Proposition 1, the following inequality holds:
0 \leq (1 - \beta )\alpha - \beta \alpha
2
\bigl[
(1 - \beta )\rho - 1 + \beta \rho - 1
\bigr]
+
+
1 - (1 - \beta )\alpha + \beta \alpha
2
-
\biggl[
\alpha
2
B (\rho , \alpha ) +
\alpha
2 (\alpha + \rho - 1)
\biggr]
\leq
\leq (\rho - 1)
\biggl\{
2\alpha - 1
2\alpha +1 (\alpha + 1)
- \beta
2
\bigl[
(1 - \beta )\alpha - \beta \alpha
\bigr] \biggr\}
.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
424 S.-R. HWANG, S.-Y. YEH, K.-L. TSENG
Proposition 3. Let \rho \geq 3, 0 \leq \beta <
1
2
, c = \beta and d = 1 - \beta in Corollary 3. Then, on the basis
of Proposition 1, the following inequality holds:
0 \leq \alpha
2
B(\rho , \alpha ) +
\alpha
2(\alpha + \rho - 1)
-
- (1 - \beta )\alpha - \beta \alpha
2\rho - 1
- 1 - (1 - \beta )\alpha + \beta \alpha
2
\bigl[
(1 - \beta )\rho - 1 + \beta \rho - 1
\bigr]
\leq
\leq (\rho - 1)
\biggl\{
\beta
2
+
\biggl(
1
4
- \beta
2
\biggr) \bigl[
(1 - \beta )\alpha - \beta \alpha
\bigr]
- 2\alpha - 1
2\alpha +1(\alpha + 1)
\biggr\}
.
References
1. M. Alomari, M. Darus, On the Hadamard’s inequality for log-convex functions on the coordinates, J. Inequal. and
Appl., Article ID 283147 (2009), 13 p.
2. S. S. Dragomir, Two mappings in connection to Hadamard’s inequalities, J. Math. Anal. and Appl., 167, 49 – 56
(1992).
3. S. S. Dragomir, On the Hadamard’s inequality for convex on the coordinates in a rectangle from the plane, Taiwanese
J. Math., 5, № 4, 775 – 788 (2001).
4. S. S. Dragomir, R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real
numbers and to trapezoidal formula, Appl. Math. Lett., 11, № 5, 91 – 95 (1998).
5. S. S. Dragomir, Y.-J. Cho, S.-S. Kim, Inequalities of Hadamard’s type for Lipschitzian mappings and their applications,
J. Math. Anal. and Appl., 245, 489 – 501 (2000).
6. L. Fejér, Über die Fourierreihen, II, Math. Naturwiss. Anz Ungar. Akad. Wiss., 24, 369 – 390 (1906) (in Hungarian).
7. J. Hadamard, Étude sur les propriétés des fonctions entières en particulier d’une fonction considérée par Riemann,
J. Math. Pures et Appl., 58, 171 – 215 (1893).
8. S.-R. Hwang, K.-L. Tseng, K.-C. Hsu, New inequalities for fractional integrals and their applications, Turkish J.
Math., 40, 471 – 486 (2016).
9. S.-R. Hwang, K.-L. Tseng, New Hermite – Hadamard-type inequalities for fractional integrals and their applications,
Rev. R. Acad. Cienc. Exactas Fı́s. Nat. Ser. A. Mat., 112, 1211 – 1223 (2018).
10. S.-R. Hwang, K.-C. Hsu, K.-L. Tseng, Hadamard-type inequalities for Lipschitzian functions in one and two variables
with their applications, J. Math. Anal. and Appl., 405, 546 – 554 (2013).
11. S.-R. Hwang, S.-Y. Yeh, K.-L. Tseng, Refinements and similar extensions of Hermite – Hadamard inequality for
fractional integrals and their applications, Appl. Math. and Comput., 249, 103 – 113 (2014).
12. U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to
midpoint formula, Appl. Math. and Comput., 147, 137 – 146 (2004).
13. U. S. Kirmaci, M. E. Özdemir, On some inequalities for differentiable mappings and applications to special means
of real numbers and to midpoint formula, Appl. Math. and Comput., 153, 361 – 368 (2004).
14. M. Z. Sarikaya, E. Set, H. Yaldiz, N. Başak, Hermite – Hadamard’s inequalities for fractional integrals and related
fractional inequalities, Math. and Comput. Model., 57, 2403 – 2407 (2013).
15. K.-L. Tseng, S.-R. Hwang, S. S. Dragomir, Fejér-type inequalities (I), J. Inequal. and Appl., Article ID 531976
(2010), 7 p.
16. G.-S. Yang, K.-L. Tseng, On certain integral inequalities related to Hermite – Hadamard inequalities, J. Math. Anal.
and Appl., 239, 180 – 187 (1999).
17. G.-S. Yang, K.-L. Tseng, Inequalities of Hadamard’s type for Lipschitzian mappings, J. Math. Anal. and Appl., 260,
230 – 238 (2001).
18. C. Zhu, M. Fečkan, J.-R. Wang, Fractional integral inequalities for differentiable convex mappings and applications
to special means and a midpoint formula, J. Amer. Math. Soc., 8, № 2, 21 – 28 (2012).
Received 11.01.17,
after revision — 08.08.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
|
| id | umjimathkievua-article-6043 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:45Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b0/384f256dcc80b5c69bccfaf38485eeb0.pdf |
| spelling | umjimathkievua-article-60432022-03-26T11:01:30Z On some Hermite–Hadamard inequalities for fractional integrals and their applications Про деякi нерiвностi Ермiта – Адамара для дробових iнтегралiв та їх застосування Про деякi нерiвностi Ермiта – Адамара для дробових iнтегралiв та їх застосування Hwang, S.-R. Yeh, S.-Y. Tseng, K.-L. Hwang, S.-R. Yeh, S.-Y. Tseng, K.-L. UDC 517.5 We establish some new extensions of Hermite\,--\,Hadamard&nbsp;inequality for fractional integrals and present several applications for the&nbsp;Beta function. УДК 517.5 Встановлено деякі нові розширення нерівності Ерміта\,--\,Адамара для дробових інтегралів та запропоновано кілька застосувань для бета-функції.&nbsp; Institute of Mathematics, NAS of Ukraine 2020-03-28 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6043 10.37863/umzh.v72i3.6043 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 3 (2020); 407-424 Український математичний журнал; Том 72 № 3 (2020); 407-424 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6043/8671 |
| spellingShingle | Hwang, S.-R. Yeh, S.-Y. Tseng, K.-L. Hwang, S.-R. Yeh, S.-Y. Tseng, K.-L. On some Hermite–Hadamard inequalities for fractional integrals and their applications |
| title | On some Hermite–Hadamard inequalities for fractional integrals and their applications |
| title_alt | Про деякi нерiвностi Ермiта – Адамара для дробових iнтегралiв та їх застосування Про деякi нерiвностi Ермiта – Адамара для дробових iнтегралiв та їх застосування |
| title_full | On some Hermite–Hadamard inequalities for fractional integrals and their applications |
| title_fullStr | On some Hermite–Hadamard inequalities for fractional integrals and their applications |
| title_full_unstemmed | On some Hermite–Hadamard inequalities for fractional integrals and their applications |
| title_short | On some Hermite–Hadamard inequalities for fractional integrals and their applications |
| title_sort | on some hermite–hadamard inequalities for fractional integrals and their applications |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6043 |
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